Local time for run and tumble particle

We investigate the local time T-loc, statistics for a run and tumble particle (RTP) in one dimension, which is the quintessential model for the motion of bacteria. In random walk literature, the RTP dynamics is studied as the persistent Brownian motion. We consider the inhomogeneous version of this model where the inhomogeneity is introduced by considering the position-dependent rate of the form R(x) = gamma vertical bar x vertical bar(alpha)/l(alpha) with alpha >= 0. For alpha = 0, we derive the probability distribution of T-loc exactly, which is expressed as a series of delta functions in which the coefficients can be interpreted as the probability of multiple revisits of the RTP to the origin starting from the origin. For general alpha, we show that the typical fluctuations of T-loc scale with time as T-loc similar to t(1+alpha/2+alpha) for large t and their probability distribution possesses a scaling behavior described by a scaling function which we have computed analytically. Second, we study the statistics of T-loc until the RTP makes a first passage to x = M (>0). In this case, we also show that the probability distribution can be expressed as a series sum of delta functions for all values of alpha (>= 0) with coefficients originating from appropriate exit problems. All our analytical findings are supported with numerical simulations.

Automated summary

This study investigates the statistics of local time T-loc for a run and tumble particle in one dimension, revealing the probability distribution and scaling behavior under different parameter conditions. The results show that the model exhibits specific scaling properties and fluctuation patterns.